In recent years, receivers based on sophisticated electronic processing techniques have received much attention in the design of high-speed optical fibre communication systems (40 Gb/s and more). In a linear regime, group velocity dispersion (GVD) and polarization mode dispersion (PMD) are the most severe sources of signal distortion and system penalty. Although the effects of GVD can be compensated by means of dispersion compensating fibres, it is known that tolerance to GVD decreases as the square of the bit rate. Hence, a compensation that would be adequate for 10 Gb/s systems might not be sufficient when upgrading to a higher bit rate because of a non-negligible residual dispersion.
Moreover, the increased sensitivity to engineering tolerances of higher transmission capacity networks can lead to unpredictable and often variable effects on the signal due to residual GVD which, in addition, can combine with the PMD, an intrinsically stochastic phenomenon. In a first order approximation the effect of PMD is considered as a differential group delay (DGD) Δτ between the two principal states of polarization (PSP) of the fibre, resulting in Inter Symbol Interference (ISI).
Usually the PMD is described by a vector {right arrow over (Ω)}, which, in a first-order approximation is assumed to be independent of frequency. Higher order effects arise when the PMD vector {right arrow over (Ω)} is frequency dependent. In a common second order approximation, {right arrow over (Ω)} is assumed to be a linear function of the frequency, {right arrow over (Ω)}={right arrow over (Ω)}0+{right arrow over (Ω)}1(ω−ω0), where {right arrow over (Ω)}1 is the derivative of {right arrow over (Ω)} evaluated at the carrier frequency ω0. Second order effects are mainly signal distortion and broadening. It has been demonstrated that with different optical compensation techniques such as, for example, a cascade of polarization controllers and polarization maintaining fibres, planar wave guide circuits or other optical devices, it is possible to recover heavy penalties caused by first or second order effects.
The techniques mentioned above, whilst effective, are often impractical because of their cost due to the use of advanced optical technologies. As a consequence much effort has been devoted to apply classical or innovative electrical equalization methods to the case of optical fibre communication systems.
One of the first electrical equalization techniques proposed for optical systems is a Feed Forward Equalizer (FFE) whose purpose is to combat the ISI induced by chromatic dispersion. Non-linear cancellation has also been postulated, since the photo detection process implies a non-linear transformation of the signal. More recently, comparisons between these compensation methods and optical compensation techniques have been presented, showing the benefits and disadvantages of both solutions.
In addition to FFE equalization and decision feedback equalization (DFE) interest is growing for Maximum Likelihood Sequence Estimation (MLSE), realized through the Viterbi algorithm (VA) by virtue of its potentially optimal performance.
In the early Nineties MLSE receivers based on the Viterbi algorithm were proposed for optical fibre systems which did not include the presence of optical amplification. Consequently, the amplified spontaneous emission (ASE) noise introduced by optical amplifiers was not taken into consideration and the statistics of the received signal, required to calculate the branch metrics of the Viterbi algorithm (VA), were assumed to be Gaussian since they are caused by the thermal and shot noise generated after the photo detection process.
In current optical systems optical amplifiers are widely used, hence the signal in the fibre is affected by noise that in the linear regime, can be modelled as additive white Gaussian noise (AWGN). However since the photo detection process performs the action of a square law detector the post detection noise statistic changes and cannot be considered Gaussian any longer. Hence, in the case of MLSE, assuming Gaussian statistics for noise after photo detection is neither realistic nor correct and leads to inaccurate results.
Accordingly, adaptive electric compensation techniques of the PMD based on the MLSE criterion have been proposed in which the statistics of the received signal are measured and updated in real time during transmission using the detected symbols and assuming no decision errors. This method, which assumes specific constraints such as, for example, sample quantization, memory length, filter type and parameters, or even the absence of filtering, has been compared with classical equalization schemes, shows an improved performance.
We have realised that it would be desirable to provide an expression of the VA branch metrics which implements the MLSE criterion for realistic values of the system parameters, whether by sampling the signal with a period equal to the symbol time or at higher rates, given that the oversampling ensures obtaining sufficient statistics of the signal received.
In particular, through numerical evaluation in accordance with a preferred embodiment of the invention, a practically exact expression of the signal statistics is derived in the case of a receiver working at a rate equal to the symbol time.
In the case of oversampling, however, at present there is no expression (exact or approximate) for the statistics of the samples. It is however possible to recur to an adaptive type receiver based on histograms. In accordance with a preferred embodiment of the invention, a method is specified based on an approximate expression in closed form of the VA metrics, which entails a negligible loss of performance compared with an optimal expression.